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Mobile Communications Handbook

Jerry D. Gibson

Baseband Signaling and Pulse Shaping

Publication details https://www.routledgehandbooks.com/doi/10.1201/b12494-5 Michael L. Honig, Melbourne Barton Published online on: 21 Aug 2012

How to cite :- Michael L. Honig, Melbourne Barton. 21 Aug 2012, Baseband Signaling and Pulse Shaping from: Mobile Communications Handbook CRC Press Accessed on: 25 Sep 2021 https://www.routledgehandbooks.com/doi/10.1201/b12494-5

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The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The publisher shall not be liable for an loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. Downloaded By: 10.3.98.104 At: 10:19 25 Sep 2021; For: 9781439817247, chapter3, 10.1201/b12494-5 2 (vectors) blocks of sequential into grouped are { bits source of sequence The system. communication a of diagram block simple 3.1aa shows Figure Communication System Model 3.1 bandwidth. finite linear with as invariant time and characterized be can that channels for pulse a such selecting in involved principles basic the discuss we chapter, this In pulse. continuous-time particular a to mapped is sequence this from taken bits of group successive each general, In transmitter. the at waveform continuous-time a to converted be must signal, analog digitized a or data representing bits, source of sequence a Consequently, input. waveform accept a as such continuous-time as channels, radio channels, communication Many physical Barton Melbourne Honig L. Michael m pulses, pulses, The transmitted signal as a function of time can be written as written be can time of function as a signal transmitted The p ( b i ; t ), which is transmitted over the channel. over the transmitted is ), which Further Reading References 3.7 3.6 3.5 3.4 3.3 3.2 3.1 Examples Considerations Additional Signaling Partial-Response Eye Diagrams Filtering Matched with Criterion Nyquist Criterion Nyquist the and Interference Intersymbol Model System Communication Long-Term Evolution (LTE) (WiMAX) Access forMicrowave Interoperability Worldwide CDMA2000 (W-CDMA) Access Multiple Division Code Wideband Complexity PowerAverage Constraints Spectral Power and Transmitted Average Precoding Eye Slope of the Inner Eye Opening Vertical Pulse Cosine Raised Baseband Signaling and t s ) ( ...... − = m . ∑ ...... • i bits { bits • Interference to Tolerance (WLAN) Networks Area Local Wireless t p • ...... ; ( Channel and Receiver Characteristics • Characteristics Receiver and Channel b i b i }, and each binary vector vector binary }, each and • Eye Opening Horizontal iT )

...... Pulse Shaping ...... • Privacy and Security and Privacy ...... • b i is mapped to one of mapped is • • Peak-to- •

...... • 3 (3.1) 35 b 44 42 54 54 50 45 38 35 52 i } Downloaded By: 10.3.98.104 At: 10:19 25 Sep 2021; For: 9781439817247, chapter3, 10.1201/b12494-5 mapped to a sequence of pulse shapes. (b) Channel model consisting of a linear, time-invariant system (transfer (transfer system time-invariant linear, a of consisting model Channel (b) shapes. pulse of sequence a to mapped 3.1 FIGURE 36 transmitted signal as signal transmitted the pulse a to bit 0 a pulse. transmitted toabaseband mapped is bits of source group counterparts. (passband) modulated however, applies; still 3.1b in and model 3.1a, the case, Figures that In receiver. the at demodulated and frequency carrier appropriate an by lated be modu signal transmitted the that requires condition this In general, spectrum. transmitted the with around centered is content frequency constant. remain parameters multipath the which during periods time as such over be model considered can valid short this Nevertheless, multipath. impairments, duerapid fading to time-varying time-varying for account not does It multipath. time-invariant and ­bandwidth where as ­filter,denoted andto band frequency transmitted the of outside noise response. frequency channel remove for the ­compensate to is filter receiver the of purpose a and filter front-end a The contains strategies. it detection of that variety wide for a valid is assumption only 3.1a. Figure This in ­sampler, shown as assume to sufficient is it discussion, this of purpose the however, for complicated; quite be can receiver The bits. source the as of denoted estimates determine is to receiver channel the of output The noise. background additive as as (due well Foritways. dispersion and mayexample, pulse multipath, introduce bandwidth) to finite therefore (bit) is rate ­information 1/ where noise. by additive followed function) As a simple example of baseband signaling, we can take take we can signaling, As a ofsimple example baseband are signals all that assumed is it 3.1b, and 3.1a, Figures In finite as such impairments, channel time-invariant linear, all for accounts model channel This time-invariant linear, a of consists and 3.1b Figure in shown is model channel used commonly A signal input the distort may which link, radio a be 3.1acan Figure in channel The rectangular g ( t T ) is the channel impulse response associated with with associated response impulse channel the ) is s h rt a wih ah ru of group each which at rate the is (b) (a) a Cmuiain ytm oe. h suc bt ae rue it bnr vcos wih are which vectors, binary into grouped are bits source The model. system Communication (a) pulse given by given pulse p Bits G {b ( ( i t } ), and a 1 bit to the pulse – pulse the to bit 1 a and ), s(t) f ), followed by additive noise noise ), by followed additive Serial-to-parallel G( f) p ( t m ) = / T

{b Baseband signaling Baseband t g 1, 0 1, . ) ( f i

} t x = ) (

= ∗ < 0 (DC). The channel passband, therefore, partially coincides coincides partially therefore, passband, channel The (DC). 0 Select pulse

t s t n(t) = ) (

p(b + ≤ baseband-equivalent ( [ p

T t g i ( ; t) t , and and , m ). Perhaps the simplest example of a baseband pulse is is pulse baseband a of example simplest the ).Perhaps x(t) −∞ n ∫ ( ) ∞ bits, or pulses, is introduced to the channel. The The channel. the to introduced is pulses, or bits, ( ∗ t t g ). The channel output is, therefore, is, output ).channel The t s ) ( p s(t) ( )] − t and and ) τ τ + = m Channel G s

t n 0 elsewhere. In this case, we can write the the write can we case, this In elsewhere. 0

) ( ( = ) ( pulse shaping pulse f

baseband signals baseband 1 (map bitsource each to a pulse), assign Mobile Communications Handbook Communications Mobile ), and the asterisk denotes convolution, denotes asterisk the ), and d

τ signals must be derived from their their from derived be must signals x(t) x Receiver ( t filter ), which is processed by the the by processed is which ), refers to the way in which a which in way the to refers , which means that the the that means which , y(t) s iT ( t ) in a variety of variety a in ) {y i } (3.2) - Downloaded By: 10.3.98.104 At: 10:19 25 Sep 2021; For: 9781439817247, chapter3, 10.1201/b12494-5 pulses may pulses have spectrum as a many the within period, symbol as transmitted transitions three are pulses of These 1/2. multiples For example, letting increase 1/ratesymbolthe as same the is case this in binary valued, and they amplitude modulate the transmitted pulse rate symbol symbol each where Shaping Pulse and Signaling Baseband 3.2 FIGURE where out that the discussion and results chapter in this apply to complex-valued symbols and pulse shapes. notconcerned relationthe with betweenpassbandandbaseband the equivalent models andsimply point whichcentered is frequency around carrier spectrum (That is, transmitted generatinga a where be Another example of a set of orthogonal pulse shapes for shapes pulse ofof orthogonal a set example Another binary called is shapes choice of pulse This The preceding example is called As an example of a signaling technique which is not PAM, is let which technique of asignaling example an As In general, the transmitted symbols { complex valued f j 1 and and m − = and choose , namely, the rate at which the symbols symbols the at which rate , namely, the f Four orthogonal spread-spectrum pulse shapes. pulse spread-spectrum orthogonal Four 1 2

. This is a consequence of considering the baseband equivalent of passband modulation. passband of equivalent baseband the considering consequence of a is This . ≠

f . For example, each successive pair of bits might select a symbol from the set {1, –1, 1 are fixed frequencies selected so that that so selected frequencies fixed are m A i p (t) p (t) A takes on a value of value a on takes

3 –1 1 = −1 1 1 i

from one of 2, each pair of bits can be mapped to a pulse in the set { orthogonal binary pulse amplitude modulation (PAM) t p t p ; ( ; ( M 1 0

= A ) )

i 2 = }, the baseband pulse = t s m ∫ + ) ( T 0 values to transmit at bit rate      1 or –1, depending on the value of the the of value the on –1,or 1 depending      t p 0 T 0 , namely, ; ( T T − = 0 1 . As a simple extension of this signaling technique,signalingsimple can extensionweathis ofAs . 2 2 frequency-shift keying (FSK) keying frequency-shift 2 2 sin( ∑ sin( ( ) i t p t t A p A ) ; π i π i are introduced to the channel. tothe introduced are t f t f 2 ) ( 1 d p (t) p (t) i t ) ) t −1 −1 4 2 f 1 1 1 = T m 0 e 0 elsewhere T and and 0 lsewhere

< < < p =

(

p T t T t t 2 bits/ ), and channel impulse response ( m < t f ). Theinformation rate(bits per second) 2 T

= (number of cycles for each bit) are bit)are each for cycles of (number

1 and

T m is shown in Figure 3.2. As these these As 3.2. Figure in shown is / T . This isknown as . , since the data symbols p ( t ), – T T i p th bit, and 1/ and bit, th ( t t t ), 3 p f ( c t .)Here we are ), –3 M -ary PAM. p occupies ­occupies ( T t g )}. ( is the the is t A ) can j (3.3) (3.4) , – i are 37 j }, Downloaded By: 10.3.98.104 At: 10:19 25 Sep 2021; For: 9781439817247, chapter3, 10.1201/b12494-5 * if choose we Specifically, interference. symbol. desired of each detection the with interfere symbols right-hand side of Equation 3.6 is called 1/ by samples received the scaling by error without bits source the obtain at response impulse ­system filter outputof the at the where then is filter function transfer g by Equation 3.3 where where 3.3 byEquation { of levels sequence 3.3. Thesource bits { Figure in of a PAMillustrated signal transmission the Consider theNyquist and Intersymbol Interference Criterion 3.2 signals signals.* narrowband reliable are transmitted than other forfrom interference to required respect with robust smallest more are the signals Spread-spectrum than 2/ of band rate The data larger shape. a much pulse assuming a ­transmission, rectangular across a spread with PAM therefore, binary is, of ­spectrum spectrum transmitted the times four roughly 38 3.3 FIGURE iT filter 3.3. inputThetransmitter the Figure to

( This example can also be viewed as coded binary PAM. Namely, each pair of two source bits are mapped to 4 coded bits, bits, tocoded 4 mapped are bits source two of pair each Namely, PAM. binary coded as viewed be also can example This which are transmitted via binary PAM with a rectangular pulse. arectangular PAM with binary via transmitted are which t ). The channel is represented by the transfer function function transfer the by represented is channel The ). ), and the receiver filter has transfer function function transfer has filter receiver the ), and One possible criterion for choosing the transmitter and receiver filters is to minimize intersymbol intersymbol minimize to is filters receiver and transmitter the choosing for criterion possible One the is 3.6 Equation of side right-hand the on term first The Let h ñ ( ( t t ) be the overall impulse response of the combined transmitter, channel, and ) receiver,ofresponse channel, be impulse the combined the which overall transmitter, has ) = Bits {b

i r } ( Baseband model of a pulse amplitude modulation system. modulation amplitude of apulse model Baseband t )* Select level n ( H t A ) is the output of the filter output ) ofis filter the the ( i }, which modulate the transmitter pulse pulse transmitter the }, modulate which f ) p ( = {A t R ) is the impulse response of the transmitter transmitter ofthe response impulse the is )

P i ( } ( f t f

) at the symbol rate 1/ rate symbol ) at the = Transmitter filter ) k y G

) ( 0. If this were the only term on the right side of Equation 3.6, we could could we 3.6, Equation of side right the on term only the were this If 0. ( A T f ) P( R + = − = ( f t y p

f ∑ h A ) intersymbol intersymbol interference ) ( ( ). We can write write can We ). i t k ) and ) and k h T − = ) ( ) ( 0 . This type of signaling is referred to as to is referred signaling of type This . i k h ∑ P T R ) ( s(t) i ( r ( f ( i T ∑ R h A f t k i = ) is the modulated sequence of delta functions of sequence functions delta ) modulated is the ≠ ) so that ) so ( ) with input ) with Channel i      T f G( ) ( h A 1 ) with associated impulse response response impulse associated ) with 0 , we can write the the write , we can i t n T i f ) ) ( k k kT + G n T h ≠ = ( ( t ) ( + − f ) + 0 0 kT n(t) ) (plus noise), which has impulse response response impulse has noise),(pluswhich ) = iT + p

n

( , which thereflects view that ) ( p Mobile Communications Handbook Communications Mobile t ( t ). The channel input is, therefore, given therefore, input is, ).channel The t ( x(t) k ). Assuming that samples are collected collected are samples that ). Assuming t )* th transmitted symbol scaled by the the by scaled symbol transmitted th

k n g ) ( ( Receiver filter t pulse-shaping filter P filter pulse-shaping )* k T th sample of sample th r ( R( t

). The output of the receiver receiver the of output The ). h f ) (0). The second term on the on term second (0). The y(t) b y i ( } are mapped to } mapped a are spread-spectrum t ) as iT r ( {y t ). ( i } neighboring ­neighboring f ) shown in in shown ) ∑ i A (3.6) (3.5) (3.7) i δ ( t – . Downloaded By: 10.3.98.104 At: 10:19 25 Sep 2021; For: 9781439817247, chapter3, 10.1201/b12494-5 suppressed. It is instructive, however, to examine the properties of the zero-forcing solution. zero-forcing of the properties the however, toexamine It instructive, is suppressed. intersymbol when increases generally which intensity, noise the on depends also error of probability the because is This desirable. be not may solution zero-forcing a used, scheme detection zero-forcing the then Shaping Pulse and Signaling Baseband 3.4 FIGURE function son, { { samples symbols data input of sequence the relates that function transfer the the to where where transform, transform Fourier inverse the as written fore, be channel where We now view Equation 3.7 in the frequency domain. Since Since domain. frequency the 3.7We in Equation now view of choice This eliminated. been has interference intersymbol the case, this In This relation states that that states relation This Fourier discrete an inverse as rewritten be can integral this manipulations, of series a Through H aliased P h eq ( k ( G k

z f = H y th received sample is sample received th ( ) is called the the called is ) ) is the Fourier transform of transform Fourier the is )

i }, where where }, ( h f ouin sne t ocs h itrybl nefrne o eo Dpnig n h tp of type the on Depending zero. to interference intersymbol the forces it since solution, ). We will assume that that assume We).will f ( Equivalent discrete-time channel for the PAM system shown in Figure 3.3 [ 3.3 Figure in shown PAM for system the channel discrete-time Equivalent kT frequency response response frequency ). ). Sampling the impulse response response impulse the Sampling ). y i

=

y H ( equivalent discrete-time transfer function transfer discrete-time equivalent iT eq H ), in the absence of noise. This is illustrated in Figure 3.4. For this rea this For 3.4. Figure in illustrated is This noise. of absence the in ), ) ( e eq j 2 ( π z fT ), ), k h H ) ( G z + = = eq

T T ( = k h (e T f T {A

) ( 1 1 p e f H ) j2 = j2 ( ∑ i ∑ H T k y = π } t ) ( π ), the bandwidth of bandwidth the ), k f k ) (

f T 0, | 0, T ). From Equations 3.10 and 3.6 we conclude that that conclude we 3.6 and 3.10 Equations From ). − , is the discrete Fourier transform of the sequence { sequence the of transform Fourier discrete the is , = 2 1 H A T f P f H 2 1 /( /( ∫ = eq    − f W    T (z) ∫ W | T + = ) f P ) > H ) ( ) (

+ W ) ( eq h k T k f f T f G . The sampled impulse response response impulse sampled The . ( k ) ( t    e e ) therefore changes the transfer function function transfer the changes therefore ) d e    j j {n f G k n 2 2 j2 ˜ π π + ) ( i π    fT } f R fk T T ) ( { y + h i

} H ( T

k t fk ) has Fourier transform Fourier ) has ( T    f d f R ) is limited by the bandwidth of the the of bandwidth the by limited is ) f   

for the overall system transfer transfer system overall the for + A i } to the sequence of received received of sequence the } to T k   

p y ( t i

) and and ) =

y h ( iT ( kT interference is is ­interference ), r ( ñ ) can, there can, ) t ) is called a called is ) i

=

ñ H ( iT (3.10b) (3.10a) eq H )]. ( (3.8) (3.9) z ( h ) is is ) 39 k f }, - - ) Downloaded By: 10.3.98.104 At: 10:19 25 Sep 2021; For: 9781439817247, chapter3, 10.1201/b12494-5 observations. condition frequency-domain (3.7) tothe condition equivalent is 40 3.5 FIGURE This relation is called the called is relation This Since 2. 4. 3. 1. For the minimum bandwidth bandwidth minimum the For bandwidth channel the criterion, Nyquist the satisfy To The pulse shape shape The pulse that states which is, That bandwidth. Nyquist the twice to bandlimited is channel the that Suppose (Since time-varying) channel. time-varying) shape pulse the Typically, channel. either functions showswhich transfer different two 3.13 Equation as 3.3). rewrite We then can of tion | band Nyquist [–1/(2band frequency The pulse. Nyquist variance bol | for G f f | | ( H Assume for the moment that momentthat the for Assume f > < f

+ in the same interval. same the in eq

1/ 1/(2

(e n P ∫ T f / j2 Two examples of frequency responses that satisfy the Nyquist criterion. Nyquist the satisfy that responses Two of frequency examples ( [ −∞ T . The condition condition . The (3.11)becomes then ∞ π T f H f ) T t h ) and ) and ) or ) ) is the discrete Fourier transform of the sequence { sequence the of transform Fourier discrete the is ) ( = 2 ) ( f

E 0 for 0 . ) [| R = d A ( H p

T t H H f ( i ( | t f = ) can be fixed, and the other filter can be adjusted or adapted to the particular particular the to adapted or adjusted be can filter other the and fixed, be can ) 2 (– ) enters into Equation 3.11 only through the product product the 3.11 Equation ) enters into through only ( − in some interval of positive length for all all for length positive of interval some in f ].) The impulse response in Equation 3.12 is called a called is 3.12 Equation in response impulse ].)The ) f 2 f 1 ) must have odd symmetry about symmetry havemustodd ) T = )]. This is the case when the receiver filter is the matched filter (see Section Section (see filter matched the is filter receiver the when case the is )].This , the transmitted signal signal transmitted the Nyquist criterion Nyquist

0 elsewhere. This implies that the system impulse response is given by is response impulse system the that implies This 0 elsewhere. f H f H ) ( ) ( H( f ) − + W H − + 2 ( H

f H 1 T = f p   

) and and )    t h 1/(2 ( T t 1 ) ( H ) is fixed, and the receiver filter is adapted to the (possibly (possibly the to adapted is filter receiver the and fixed, is ) . From Equations 3.10b and 3.11 we make the following following the make 3.11we and 3.10b Equations From . eq T = f T ) ( h ), Equations 3.10b and 3.11 imply that that imply 3.11 and 3.10b Equations ), T 1 T f e ), 1/(2), ( sin(    j t    2 H ) are both real valued, so that that so valued, real both are ) π < = fT π t S + + ( ) ( T t π f f H / ) that satisfy the Nyquist the criterion. )satisfy that T t T = , ) /    − = )] [that is, the passband of passband the is, )][that 1 0

∑ −

2 f 1

T T i 1 = Mobile Communications Handbook Communications Mobile f h A

W    1/(2 i < = must be at least 1/(2 least at be must ) ( i t 2 n T 1 T T h , which implies that that implies which , ). This is illustrated in Figure 3.5, Figure 3.5, in illustrated is ).This H k

}, the time-domain, or sequence sequence or time-domain, },the

( T f ) has power equal power equal has 2 1 T P minimum bandwidth minimum ( f H ) R ( H ( f ) is an even func even an is ) f ( ). Consequently, f T )] is called the the called )]is H ). Otherwise, Otherwise, ). H G eq to the sym to the ( ( (e f f ) ) j2 = π = f

T (3.12) (3.13)

(3.14) T (3.11) 0 for 0 ) for =

or

0 - - Downloaded By: 10.3.98.104 At: 10:19 25 Sep 2021; For: 9781439817247, chapter3, 10.1201/b12494-5 of infinite duration. They can be approximated in practice by introducing an appropriate delay, and delay, appropriate an introducing by practice in approximated be can They duration. infinite of (2 where givenused Equation by be 3.12 should pulses efficiency, Nyquist bandwidth To maximize function transfer with ideal is channel the that Suppose Pulse Cosine Raised 3.2.1 Shaping Pulse and Signaling Baseband H that 0 where transform Fourier has which the is technique, first pulse, the a of illustrates which example common most The 3.5. Section in approach latter the examine will We efficiency. impractical. therefore is signaling bandwidth Minimum { symbols offset jitter. is to not Namely,respect sampling of robust with sampling a signaling small type this that fact the is important, is more which drawback, second The closely. approximate to difficult 3.15 is Equation by given response frequency bandlimited ideal the that observing to equivalent is This wide. be must dow slowly, as however,namely, 1/ very decays The pulse, pulse. the truncating ( T Plots of Plots power sacrifices approach latter the whereas efficiency, bandwidth sacrifices approach former The in practice: ways two in one of solved is generally problem The preceding 1/ as decays pulse Nyquist the Since f 2. ). This type of signaling, however, has two major drawbacks. First, Nyquist pulses are noncausal and are noncausal pulses Nyquist First, however,twomajordrawbacks. has of signaling, type ). This 1. i te yus ple ih iiu bnwdh 1/(2 bandwidth minimum with pulse Nyquist the is ) h ( ε A The pulse bandwidth is increased to provide a faster pulse decay than 1/ than decay pulse faster a to provide increased is bandwidth The pulse subtracted out at the receiver. out at the subtracted t produces the outputsample the produces ) satisfies the) criterionNyquist satisfies (3.7)and,consequently, ≤ controlled A

α p i } can, therefore, lead to very large intersymbol interference, no matter how small the offset. offset. the howsmall matter no interference, intersymbol large very to lead therefore, can, }

( ≤ t ) and and )

1. amount of intersymbol interference is introduced at the transmitter, which can be be can which transmitter, the at introduced is interference intersymbol of amount P f H ( f ) ( ) are shown in Figures 3.6a and 3.6b for different values of values different for 3.6b and 3.6a Figures in shown are ) + =          T 0 T T 2      k y 1 ) ( t h ) ( A T co = + t = s , this sum is not guaranteed to converge. A particular choice of choice particular A converge. to guaranteed not is sum this ,    ε f G    π α raised cosine pulse cosine raised sin( ) ( π    ∑ T t π = i / f T t /      − 1 0 i , c ) , 1 sin[    2 −    ε π T 2 1 W f W f α ) ( ε π i k − os ) (    > < i k + − ) (    ) ( απ      + − T α , given by , given ), and when when and ), T t H T t

/ 0 1 / ( f 2 / − ≤ ≤ f T T 2 / > ) satisfies Equation ) 3.11.satisfies When α T    f 1

] ≤ ≤ 2 +

T α f 1 t 2 − , so that the truncation win truncation the that so , α T α

> 1 t

2 . + 0, 0, T α H α

( . It is easily verified verified easily is It . f ) has bandwidth bandwidth has ) W α (3.17) (3.15) (3.16) (3.18)

= =

41

1/ 0, 0, - Downloaded By: 10.3.98.104 At: 10:19 25 Sep 2021; For: 9781439817247, chapter3, 10.1201/b12494-5 f G 42 sampling instant, is instant, sampling y then is filter receiver where of Consider an the transmission isolated pulse Nyquist Matched Criterion with Filtering 3.3 response frequency receiver the setting channel, ideal an R Assuming 3.18. Equation in response response frequency has the which the shape, in transmit to constant is not alternative An generally noise. is Gaussian additive response of ­presence filter receiver (matched) optimum the however, channel, (transparent) ideal an with Even constant. is response frequency channel and receiver combined the if shape pulse receiver. If transmitted the offsets. tosampling respect 1/ as decays pulse cosine raised the As bandwidth. minimum the 1/ twice bandwidth pulse the because is This bandwidth. excess 100% with pulse cosine raised a is pulse ­bandwidth (1 3.6 FIGURE (a) ( ( ) ( t + f ) is sampled at sampled is ) Thefirstterm on theright-hand side and is thetheseconddesired thatsignal, term isAssuming noise. and channel, transmitter, of combination the to (3.18) response applies frequency cosine raised The

) h(t)

–0.5 α = 1.0 0.0 0.5 = )/(2

P t g f P ( –4 ) ( f T ) ( ) then results in an overall raised cosine system response response system cosine raised overall an in results ) then as a fraction of the minimum bandwidth 1/(2 bandwidth minimum the of fraction a as Excess bandwidth ) with a raised cosine rolloff. The parameter parameter The rolloff. cosine raised a with ) is the inverse Fourier transform of the combined transmitter-channel transfer function function transfer transmitter-channel combined the of transform Fourier inverse the is f G (a) Raised cosine pulse. (b) Raised cosine spectrum. cosine (b) Raised pulse. cosine (a) Raised ) ( 100% 50% 0 . We will assume that the noise noise the that assume will We . t –2

=

0, the ratio of signal energy to noise energy, or energy, noise to energy signal of ratio the 0, t y 4 2 0 ) ( = = t r ) ( * p P ( t x ( t ) is a raised cosine pulse, then then ) pulse, cosine is a raised t x ) ( f ) ( ) given by the square-root of the raised cosine frequency frequency cosine raised the of square-root the by given ) A + = r A 0 δ g A 0 ( ( [ 0 t n ). In the case input this to the receiver in ( g t ) ( t (b) n t ( ) ) is white with spectrum spectrum with white is ) *

α H( f ) 0.0 0.5 1.0 , therefore, represents the additional, or additional, the represents therefore, , T –1.0 r t ) ( ). For example, when when example, For ). )] t +

Mobile Communications Handbook Communications Mobile ( [ n t ( ) H * –0.5 ( signal-to-noise ratio signal-to-noise f h t ). square-root raised cosine raised square-root t )] ( Excess bandwidth 3 t , performance is robust with with robust is performance , ) is a raised cosine pulse only ) pulse cosine is a raised

100% 50% 0 0.0 N 0 α /2. The output of the of output The /2.

=

1, we say that the the that say we 1, 0.5 ( Figure 3.3 SNR ) at the the at ) pulse pulse excess excess (3.20) (3.19) T 1.0 is is is Downloaded By: 10.3.98.104 At: 10:19 25 Sep 2021; For: 9781439817247, chapter3, 10.1201/b12494-5 ( | filter transmitter-channel ­combined when detecting a sequence of channel symbols with intersymbol interference (assuming the additive additive the (assuming interference intersymbol with symbols channel of sequence a detecting when Shaping Pulse and Signaling Baseband FIGURE 3.7 FIGURE spectrum. folded channel 1/(2| frequency response channel the which in operation folding a as described 3.14 where Equation by given then bandwidth, Nyquist the twice to bandlimited Note that of noise. absence { symbols transmitted of sequence the relates which pulse baseband the where is filter receiver the of output the and 3.7, given by Figure in shown is model baseband the case, this In filter. noise is Gaussian). We, therefore, reconsider the Nyquist when criterion the receiver isfilterthe matched f R t g f G ) ( ( ( − Choosing the receiver filter to be the matched filter is optimal in more general situations, such as such situations, general more in optimal is filter matched the be to filter receiver the Choosing is expression this maximizes that response impulse The receiver With a matched filter at the receiver, the equivalent discrete-time transfer function is function transfer discrete-time equivalent the the receiver, at filter amatched With ) )| ], which is known as the the as known is which ], = 2 = f G | ∗ f P ) ( ) ( ) Baseband PAM model with a matched filter at the receiver. the at filter amatched PAM with model Baseband . f G T ). For this reason, reason, this For ). {A | i 2 } Ti iple epne s the is response impulse This . T ransmitter filter H h eq P( H ( H (e t ) is now the impulse response of the filter with transfer function function transfer with filter the of response impulse the now is ) f ( eq j2 ace filter matched ) f π ) ) ( f e SN T H = ) is positive, real valued, and an even function of function even an and valued, real positive, is ) j 2 f G

eq π | R t y G fT ) ( (e ) ( t h ( = ) ( s(t) j2 f − = = π , ) − = f A E T P T + = T 1 1 ) with a matched receiver filter is often referred to as the as to referred often is filter receiver matched a with )   ( ∑ Channel ∑ −∞ f ∑ i ∫ G( ∞ mus rsos. h ascae tase fnto is function transfer associated The response. impulse )| k k 0 N s g h A 2 f 2 2 . The aliasing sum in Equation 3.10b can therefore be therefore can 3.10b Equation in sum aliasing The . ∗ ) f G f P i   0 ) ( ∫ ∫    ) (    H i t A −∞ −∞ ∞ ∞ s g ( autocorrelation k ) ( f } to the sequence of received samples { samples received of sequence the to } t r t r − n(t) ) ) ( ) ( n T + − T = T k k

s t 2    0 for | for 0    x(t) + t g d f G d 2 ) ( t    ) ( t d

Recei t f

− G*( | 2 H t r > T ) ( k (

ver f of the impulse response of the the of response impulse the of 1/ f    )P*(f )| T − = 2 2 filter is folded around the Nyquist Nyquist the around folded is , and the Nyquist condition is is condition Nyquist the and ,

t g ) ∗ ) ( y(t) [complex conjugateof iT f {y . If the channel is is channel the If . i } y k } in the the in } (3.24) (3.22) (3.23) (3.21) 43 Downloaded By: 10.3.98.104 At: 10:19 25 Sep 2021; For: 9781439817247, chapter3, 10.1201/b12494-5 nevl 0 interval response impulse the turn, of duration in the on depends This, 3.6. Equation in shown interference, intersymbol the of extent the on depends segment surrounding symbols ­channel display. oscilloscope the 1/ rate into to on translated beone symbol symbolinterval intervals This successive causes the plates. deflection horizontal at waveform sawtooth waveform a received applying and the oscilloscope applying the of by plates ­deflection oscilloscope an on viewed easily is diagram theeye In practice, levels. signal theThe transmitted isof of number eyenumber than one openings less length of segments successive receiver at the signal data 3.8a shows the Figure channel. bandlimited ideal an and 25%bandwidth shape with excess pulse cosine the is system to examine communications in duea interference digital of to One distortion intersymbol way severity the to assess Eye Diagrams 3.4 44 3.8 FIGURE at times sampling nominal ing SNR ± where 3}, where each symbol is equally likely, and and likely, equally is symbol each where 3}, (a) Each waveform segment segment waveform Each the for 3.8b in Figure shown is picture The resulting 2. 1. is 30 –5 –4 –3 –2 –1 y(t) 0 1 2 3 4 5 Translate each of these waveform segments [ segments waveform these of each Translate Partition the waveform waveform the Partition interval [– h ( t 5 0 ) is given by Equation 3.17, Equation by given is ) dB.) The eye diagram is constructed from the time-domain data signal data signal time-domain thefrom isconstructed dB.) diagram eyeThe <

t

(a) Received signal (a) signal Received <

mT T /2, , then each waveform segment depends on approximately approximately on depends segment waveform each then , T Symbol intervals /2], andsuperimpose. eye eye diagram 10 y y iT ( y A kT t ( , ( ), t ( , ) into successive segments of length of length segments successive ) into t k ). (b) Eye diagram for received signal shown in Figure 3.8a. Figure in shown signal for received ). (b) Eye diagram i . The number of channel symbols that affects a particular waveform particular a affects that symbols channel of number The . ,

k > k

= . The eye diagram is illustrated in Figures 3.8a andin 3.8a Figures 3.8b, for. is illustrated The aeyeraised diagram 1, is also possible. This would result in result would This possible. also is 1, α + t y

,1 ,...): . 1,. 0, 2, 15 ) ( 1/2) =

1/4, each symbol symbol 1/4,each − = T

∑ ≤ i

ñ t

( ≤ h A t 20 ) is bandlimited white Gaussian noise. (The received received (The noise. Gaussian white bandlimited is )

i h ( k ( y ) ( (b) t

i t ( ). For example, if if example, For). + t ), ( ), –5 –4 –3 –2 –1

3/2) y 0 1 2 3 4 5 –0.5 ( n T k t ) shown in Figure 3.8a. (Partitioning (Partitioning 3.8a. Figure in shown )

+ T A +

dpns n h priua sqec of sequence particular the on depends , i 1/2) is independently chosen from the set { set the from chosen independently is ) ( Mobile Communications Handbook Communications Mobile t T

≤

t T

≤ starting from from starting h

( ( t k ) has most of its energy in the the in energy its of most has )

+

3/2) i successive eye diagrams.) diagrams.) eye successive 0 T m y , symbols. Assuming Assuming symbols. ( k t ) as ) follows (assum as

y = t (

t 0, 1, 2, . . .] to the to .] . . 2, 1, 0, = ) to the vertical vertical the to )

T /2. T y ( to the the to t (3.25) ) into into ) 0.5 ± 1, - Downloaded By: 10.3.98.104 At: 10:19 25 Sep 2021; For: 9781439817247, chapter3, 10.1201/b12494-5 worst-case intersymbol interference, assuming that that assuming interference, intersymbol worst-case time any at openings The vertical Eye Vertical Opening 3.4.1 system. communications significant causes symbols channel afewsymbols. only spans of typically response impulse the sequence applications one only that possible very low with is sequence and occurs interference, this probability.) wireless ­intersymbol In digital current (It diagram. eye the in this implies transmission, binary that there are a total of 2 Shaping Pulse and Signaling Baseband 3.7. Equation condition Nyquist by the given response full not have the does response called is approach This receiver. the at removed be can which transmitter, the at interference intersymbol of is to a introduce possibility amount Another errors. controlled to respect timing with to robustness gain efficiency bandwidth compromise pulses cosine Raised compromised. be must efficiency power and/or bandwidth channel, bandlimited overideal an signaling ToNyquist with avoid problems associated the Signaling Partial-Response 3.5 sequence. canoffset leadonthelarge,depending tointerferencedata thattiming an term intersymbol is arbitrarily which is eye closed fordiagram all Equation 3.12shape pulse an produces bandwidth Theminimum eye signaling). box-shaped binary a (assuming produces diagram pulse rectangular a example, For opening. eye the wider the decays, pulse error.of the probability increases significantly times sampling the in of jitter amount cant signifi a case, this In increases. offset timing the as rapidly closes eye the that means slope steep very a Theslope ofthe inner eye toindicatessensitivity timing jitter orin thevariance timingoffset. Specifically, Eye Slope Inner ofthe 3.4.3 error the although tolerated, be can offset timing opening. vertical on the depend large will ­probability a that indicates opening horizontal wide eye narrow a very a Conversely, closed. is eye the where Specifically, sampling in result offset. will offset timing timing small a to that indicates ­opening sensitivity the indicates opening each of width The Eye Opening Horizontal 3.4.2 times signal levelsimplyalargedegreeofimmunitytoadditivenoise.Ingeneral, between spacings vertical wide Conversely, 1/2. to close be therefore, will, error of probability The bol. then the decisions will depend primarily on the intersymbol interference rather than on the desired sym to be open. A closed eye implies that if the estimated bits are obtained by thresholding the samples or all, signal levels disappears altogether. In that case, the eye is said to be closed. Otherwise, the eye is said is possible for the intersymbol interference to be large enough so that this vertical opening between some, The eye diagram has the following important features which measure the performance of a digital digital a of performance the measure which features important following the has diagram eye The To illustrate PR signaling, suppose that the Nyquist condition Equation 3.7 is replaced by the condition baseband the faster the general, In shape. pulse the by determined is diagram eye the of shape The kT partial-response (PR) signaling partial-response

+

t 0 , k

=

,1 ,...,where . 0, 1,2,. t t 0 except for t , – 0 . The terminology reflects the fact that the sampled system impulse impulse sampledsystem thethat thefact reflects . Theterminology ischosentomaximizetheverticaleyeopening. T /2 h ≤ k

t 0 =

t ≤

     =

T 0 1 a 0

0. This is because, as shown earlier, an arbitrarily small 0. isasThis small because, shown earlier, anarbitrarily /2, represent the separation between signal levels with with levels signal between separation the represent /2, k y ll ot ( = t ) is sampled at times times at sampled is ) m her , , waveform segments that can be 1 k

y ( t

t = ) shouldbesampledatthe

kT

+

t 0 , k

= superimposed ­superimposed

0, 1, 2, . . . . It . . . . 1, 0, 2, y (3.26) ( kT 45 ), - - Downloaded By: 10.3.98.104 At: 10:19 25 Sep 2021; For: 9781439817247, chapter3, 10.1201/b12494-5 in this case is the discrete Fourier transform of the sequence in Equation 3.26, Equation in sequence of the transform Fourier discrete the is case this in transmitted the detect to how of sequence discussion { defer and signaling PR of characteristics spectral the on focus we now For symbol. transmitted neighboring one from interference intersymbol is there that so 46 3.9 FIGURE bandwidth. excess zero with feasible is rate symbol Nyquist the at signaling that 3.12, is Equation pulse [2]. by Kretzmer, generalized Lender, by proposed first was PR Duobinary later filter. and [1], realizable by a physically ­approximated function transfer the response, H where sinc and G (a) ( ( The The The main advantage of the duobinary pulse Equation 3.29b, relative to the minimum bandwidth bandwidth minimum the to relative 3.29b, Equation pulse duobinary the of advantage main The the for 3.28 to Equation case, be satisfied, As full-response in the f |H( f )| f ) and transmitter filter filter transmitter ) and 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 ) in Figure 3.9. [Notice that that [Notice 3.9. Figure in ) 1 2 0 –1 k th received sample is then is sample received th –0.8 A x k

} in the presence of transfer interference. Theintersymbol discrete-time equivalent = Duobinary frequency response and minimum bandwidth pulse. bandwidth minimum and response frequency Duobinary –0.6

(sin Normalized frequencyfT –0.4 π x )/( –0.2 π x ). is a This pulse called P ( 0 f ) is ) is H f H 0.2 t h eq h W ) ( ) ( ( H ) ( t e ) satisfies Equation 3.26.] Unlike the ideal bandlimited frequency frequency bandlimited ideal the Unlike 3.26.] Equation satisfies ) 0.4

( = j = 2 = f

π 1/(2 ) in Equation 3.29a is continuous and is, therefore, easily easily therefore, is, and continuous is 3.29a Equation in ) fT t T      0.6 ( { 0 2 sinc A y f T = + = T k k e ). Assuming ). Assuming 2 1 T 0.8 + = − 1 / j π ∑ fT e e k duobinary t T cos( − 1 j f H [ ) 2 π + (b) fT n A    π sinc k k = –0.2 − h(t/T) f T 0.2 0.4 0.6 0.8 1.2 1.4 1 + P ( ) + 0 1 –4 / ( T pulse and pulse is shown along the with k ) ( f − ) has this minimum bandwidth implies bandwidth minimum this ) has    j T f π fT −

–3 Mobile Communications Handbook Communications Mobile < > T T cos( 1 1 / / –2 ) ( minimum π 2 2 ]} T fT ) –1

) Normalized timet/T

0 bandwidth of the channel of channel the bandwidth 1 2 3 associated ­associated 4 function ­function (3.29b) (3.29a) 5 (3.27) (3.28) 6 Downloaded By: 10.3.98.104 At: 10:19 25 Sep 2021; For: 9781439817247, chapter3, 10.1201/b12494-5 edge band Nyquist response frequency channel the where situations in priate appro is duobinary is response system overall the that so filters receiver and transmitter the Selecting errors. timing to respect with robust is it pulse, Nyquist a than rapidly moremuch decays pulse the As Shaping Pulse and Signaling Baseband FIGURE 3.10 FIGURE Nyquist rate. Nyquist trans a Like through channel former. the to coupled is signal transmitted the where channels (twisted-pair) wire response and function transfer discrete-time equivalent an has which (a) As another example of PR signaling, consider the the consider of PR signaling, example another As |H( f )| channel the when appropriate is shape pulse This 3.10. Figure in plotted are functions These is response system overall the bandwidth, excess zero With 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 1 2 0 –1 G –0.8 ( f ) is near zero at both DC ( DC both at zero near is ) –0.6 duobinary PR, modified duobinary allows minimum bandwidth signaling at the the at signaling bandwidth minimum allows duobinary modified PR, ­duobinary Modified duobinary frequency response and minimum bandwidth pulse. bandwidth minimum and response frequency duobinary Modified Normalized frequencyf –0.4 f = 1/(2 –0.2 T ). 0 t h ) ( 0.2 f H + = ) ( 0.4 t T H [ { T sinc = eq f h 0.6

     ) ( k = e e 0 / T j

j = 2 2 0) and at the Nyquist band edge. This is often the case for case the often is This edge. band Nyquist the at and 0) 2 ) ( 0.8 π fT      sin( 1 1 a 0 = − T T 1 1 − = = 1 f j π k k 2 2 modified j 2 (b) fT ll ot ] sin( π − = –1.5 –0.5 fT h(t/T) − − ( ) 0.5 1.5 –1 sinc her 0 1 –5 π G e T T f f − ( T [ k j duobinary partial response partial duobinary f 2 < –4 > ) ( ) π ) is near zero or has a rapid rolloff at the at rolloff rapid a has or zero near is )

t fT 2 1 1 / / ) (

–3 2 T T ) ] / –2

Normalized timet/T }

–1 0 1 2 3 (3.32b) (3.32a) 4 (3.30) (3.31) 47 5 - - Downloaded By: 10.3.98.104 At: 10:19 25 Sep 2021; For: 9781439817247, chapter3, 10.1201/b12494-5 } { 3.11 FIGURE 48 FIGURE 3.12 FIGURE Let { response. partial overall by the introduced interference this avoid To symbols. be detected can levels successive to transmitted propagate the ­problem, to error the cause will sample received the sample received symbol the decoded Consider If the received sample the signal Equation receiver 3.27 signaling. has duobinary with Precoding 3.5.1 strategy. detection receiver. suboptimal We asimple now describe at the used of detector type the as well as characteristics, channel on the depends PR scheme particular {0, is signaling forlevels duobinary signal of set received the Forto noise. example, may and reduce immunity detector the of complexity the increases which levels, signal received of number the how increases interference, ever,PRsignaling intersymbol introducing By advantages. performance significant offers ­potentially { shaping filter where pulse the signal, that [Note discrete). function transfer with filter (digital) discrete-time a through { levels transmitted { sequence the generated. is in that interference intersymbol additional ­however, of the because coefficients nonzero more choosing by ­generated 1 toas referred also is duobinary where σ b A 2 2 d A k ′ k A particular partial response is often identified by the polynomial the by identified often is response partial A particular We illustrate precoding forfirst PR.duobinary Thesequence of inoperationsillustrated is Figure 3.12. response channel the in nulls with coincident nulls have to spectrum transmitted the Shaping a PR pulse modulating than Rather in shown those than responses system complicated more general, In } are selected independently and are identically distributed, then the transmitted spectrum is is spectrum transmitted the then distributed, identically are and independently selected are } ( | by the operation by the P b k } denote the sequence of source bits where D e (for delay) takes the place of the usual usual the of place the (for delay) takes j π fT )| {A 2 for | {b Generation of PR signal. Generation Precoding for aPR channel. Precoding i } ± i } y 2} from which the transmitted levels { levels transmitted the which from 2} k . If an error occurs, however, then subtracting the preceding symbol estimate from from estimate symbol preceding the subtracting however,then occurs, error an If . Precoder f A | P P A < i d }. This is shown in Figure 3.11. Namely, the transmitted levels are first passed passed first are levels transmitted the Namely, 3.11. Figure in shown is This }. d k

(e (e 1/(2 − 1 j2πfT j2 , then in the absence of noise noise of absence the in then , π f T T {b ) ) can be selected to be be to selected be can ) ) and is zero for | zero is ) and i ′} + {A Select

level D i ′} P partial response. partial ( h f ( impulse train ) t precoded ), a PR signal can also be generated by filtering the sequence of bybe generated filtering also ),can a PR signal {A = Generate

i 1, | } b b k k ′ f f | ∑ | z ⊕ = PR channel k b > K = − < k in such a way as to compensate for the intersymbol intersymbol the for compensate to as way a such in 1

0

1/(2 in the the in

∈ 1/(2 D h H k

{0, 1}. Thissequence is totransformed the sequence eq T T Σ k b i (e A ), where k ) and is zero elsewhere. If the transmitted levels ) If and is transmitted the zero elsewhere. ′ A j2 ± k − z can be decoded by subtracting subtracting by decoded be can i ′δ 1 π 1} must be estimated. The performance of a of performance The estimated. be must 1} transform of the sequence { sequence the of transform f

T (t −iT { ).] The outputs of this filter form the PAM the form filter this of outputs ).]The y i P } Mobile Communications Handbook Communications Mobile d σ (e ) Threshold 2 A j2 π = (Ideal bandlimited f T ) (where the subscript subscript the (where ) A E h [| k }. This complicates detection, detection, complicates This }. P( f) k iue 3.9 Figures ] | (Estimated bits) 2 . {b ˆ i } ) and 3.10 can be be can 3.10 and h s(t) k }. For example, }.Forexample, A k − d 1 indicates indicates from the the from correctly ­correctly (3.33) - Downloaded By: 10.3.98.104 At: 10:19 25 Sep 2021; For: 9781439817247, chapter3, 10.1201/b12494-5 and combining Equations 3.33 and 3.35 gives 3.33 and Equations combining and then is symbol received { levels signal transmitted binary where Shaping Pulse and Signaling Baseband assumed by {–( is levels transmitted of set the that so signal levelisgivenby M { symbols source the that and integers are coefficients the where by PR specified is the that Suppose { bits of transmitted make –1}. Thesesymbolsaretransformedtothesequence That is, Thatis, Because of the modulo operation, each symbol ofsymbol each operation, modulo the Because channels. PR other to and PAM multilevel to extended be can technique precoding preceding The if is, That symbol-by-symbol ⊕

denotes modulo 2 addition (exclusive OR). The sequence sequence The OR). (exclusive addition 2 modulo denotes b b k k ′ b y = k k ). Intheabsenceofnoisereceivedsampleis

= 1 0

( ± 2, then then 2, ′ b = i }, bits precoded {b { { { 3.1 TABLE decisions that do not depend on previous decisions. Tabledo not that decisions. on depend decisions previous 3.1 shows a sequence ) y A b is mapped to the transmitted level is mapped to the transmitted i i i ′ }: i }: }: }: b k

=

− 0, and if if and 0, 0 1 A k A y } according to } according 0 1 1 1 Example of Precoding for Duobinary PR for Duobinary of Precoding Example b b k k k k } { ′ b + = − = i ′ y y b 2 1 1 0 k k , transmitted signal levels { levels signal , transmitted

   M b A = D H + =

k k 0, then then 0, eq 2 1 1 0 h y –    b A = k ) ( b A 1), . . ., ( ., . 1),. 2 1 k k = ∑ 1 2 k i − − K = = 0 0 1 1 b k 1 ∑ 1 ′ i = k ′ = K 1 2 b h = i loi h e 0 ,..., . set {0, in the is 1,also . − − b 0 k i ∑ k k 2 1 1 2 ′ ) (

K = 0 1 1 1 } { M ′    = i M ) ( 0 b − − A

i k ′ D h 1. Precoding, therefore, enables the detector to detector the enables therefore, Precoding, 1.    k mo ′ i k – k − viatheprecodingoperation

− +

1)} (i.e., a shifted version of the set of values of set the of version shifted a (i.e., 1)} mo 0 0 1 1

d A k b k ′ k

− d

= 1 − −

M − 0 0 2 1 –1 ( b k

r eetdfo h e 0 ,..., . . 1, {0, set the from selected are } A } {

A − − b i }, and received samples { samples }, received and 0 0 2 1 k k ′

= is mapped to the sequence of sequence the to mapped is

1). In the absence of noise, the 0 1 1 1 M 2 1 1 0 –1}. The k th transmitted y i }. (3.36) (3.34) (3.37) (3.38) (3.35) (3.39) 49 Downloaded By: 10.3.98.104 At: 10:19 25 Sep 2021; For: 9781439817247, chapter3, 10.1201/b12494-5 Considerations Additional 3.6 performance. improvements in toobtain PR signaling with used be can likelihood) receiver.(e.g.,schemes In presence the ofdetection maximum morenoise, sophisticated the that shown be it can and 50 reasons: the following for in practice important is constraint This well. power,a transmitted average on constraint a to addition In Peak-to-Average Power 3.6.2 slot.) time time are assigned the signals within contained be must transmitted response impulse the system the of if duration the (Conversely, then band. multiplexed, frequency assigned the of edges the at rolloff in filter transmitter the constraint, this Tomeet constraint. of amount the is to limit needed a Therefore, constraint bands. to adjacent frequency assigned ated by transmitters limit to imposed are spectrum ­transmitted frequencies. low than more crosstalk of particular amount a the at reduce to generated Consequently, frequency. of capacitive by function a caused as generally increases is and channels coupling a wireline as in power crosstalk transmitted example, For allowable frequency. of maximum the function constraint specifies the that mask dependent, spectral a frequency of is form the and interference take may of receivers are type neighboring this into constraints Because radiated radio), systems. crosstalk, digital or communications as interference, well of as amount loops, the limit subscriber to imposed digital (e.g., applications many In ters. transmit battery-powered use that applications low- wireless example, mobile for For desirable application. highly the is power to average according varies power transmitted average on constraint The Constraints Average Power Spectral and Transmitted 3.6.1 that selection. constraints this may influence practical additional on discussion brief a give we Here pulses. baseband selecting for In many applications, and bandwidth interference intersymbol are not the only important rcdn te ybl { symbols the Precoding In radio applications where signals are assigned different frequency bands, constraints on the the on constraints bands, frequency different assigned are signals where applications radio In 2. 3. 1.

Rapid fades can severely distort signals with high peak-to-average power. peak-to-average high with signals distort severely can fades Rapid The dynamic range of the transmitter is limited. In particular, saturation of the output amplifier amplifier output the of saturation particular, In limited. is transmitter the of range dynamic The noise for pulse-code modulated voice signals introduces amplitude-dependent amplitude-dependent introduces signals. data signals voice modulated reduce pulse-code to for used noise compander the Namely, [3]. pro network companding telephone the voice is the in applications cess wireline to pertains that example Another example. one waveform. transmitted “clip” the will The transmitted signal may be subjected to nonlinearities. Saturation of the output amplifier is amplifier output the of Saturation nonlinearities. to subjected be may signal transmitted The out-of-band power out-of-band transmitter, the pulse shaping filter generally attenuates high frequencies frequencies high attenuates generally filter shaping pulse the ­transmitter, b k k } in this manner, therefore, enables symbol-by-symbol decisions at the the at decisions symbol-by-symbol enables therefore, manner, this in } th source symbol is given by given is symbol source th generated by each transmitter, in addition to an overall average power power average overall an to addition in transmitter, each by generated y b k k + = 2 1 ( ( adjacent-channel interference adjacent-channel H M ⋅ − 1 1 ( ) eq )) peak-power

Figure 3.3 Figure mo Mobile Communications Handbook Communications Mobile d M

constraint is often imposed as imposed often is constraint must have a sufficiently steep steep sufficiently a have must . This interference is gener is interference This . considerations ­considerations distortion in in ­distortion quantization ­quantization (3.40) - - - Downloaded By: 10.3.98.104 At: 10:19 25 Sep 2021; For: 9781439817247, chapter3, 10.1201/b12494-5 signaling with rounded pulses is often used. Operating RF power amplifiers with power back-off can back-off power amplification. with PAR, reduce toinefficient but leads also amplifiers power RF Operating used. often is pulses rounded with signaling PAR. However, efficiency.compromises bandwidth Inthis where applications PAR should be low, binary where ratio power peak-to-average Shaping Pulse and Signaling Baseband efficient pulse shapes can complicate other system functions, such as timing and carrier recovery. carrier and timing as such functions, system other complicate can shapes pulse efficient bandwidth- addition, In cutoff. sharp a with filter a requires signal bandwidth-efficient ofa Generation Complexity 3.6.4 loops). subscriber digital and modems (voiceband operation. Both multitone haveand channels modulation precoding been wireline used with threshhold simple a by data the detect to receiver the allowing channel, the by introduced interference intersymbol the for compensates precoder The precoder. a in sent used and is it where receiver transmitter, the the at to back measured is response impulse channel channels. discrete-time partial-response equivalent for the earlier Namely, described technique the of extension an is precoding Adaptive response. frequency channel the to adapt can transmitter the which in way [6,7])another is ­precoding ferred to adapting the pulse transmitter shape. However, the following examples are notable exceptions. complication,of extra receiver the adaptingofoften Becausethis is pre characteristics. channeltransmitter changesin the notify can receiver the which through channel feedback a requires channel varying receiver adapt to the changing channel characteristics. Adapting the transmitter to compensate for a time- phase, may allow more bandwidth efficientpulse shapes in addition tomultilevel signaling. afast fading environment noncoherentwith detection. The ability totrack channel characteristics, suchas influencethe choice of atransmitted pulse shape. For example, a constantamplitude pulseThe istype of channelimpairments encounteredand the type of detection schemeused atthe receiver Receiver and Characteristics can Channel also 3.6.3 PAR. minimize to OFDM specifically tones certain aside setting and bols sym transmitted of set the altering include These therein. references and 4 Reference in described are approaches proposed Some OFDM. for required are reduction PAR to approaches sophisticated more Hence system. single-carrier PAR equivalent to high an a compared very exhibit can signal ­transmitted The preceding impairments or constraints indicate that the transmitted waveform should have a lowshould waveform transmitted the that indicate constraints or impairments The preceding In addition to multitone modulation, modulation, multitone to addition In and/or transmitter the that requires channels time-varying over communications data High-speed an For 2. 1. Multitone modulation divides the channel bandwidth into small subbands, and the the and subbands, small into bandwidth channel the divides modulation Multitone Mobile cellular systems may dynamically adapt the transmitter power for each link (in both both (in link each for power transmitter the adapt dynamically may systems cellular Mobile to guide the allocation of transmitted bits and power [5]. and bits of transmitted allocation the to guide the to back transmitted be must subband each for ratio signal-to-noise received The rate. information the maximize to subbands these among distributed are bits source and power varied. is shape pulse the with associated parameter ainwhich single shaping pulse transmitter as aviewed simple form beof adaptive can This tions. to ofcondi to generated and amount control forthe compensate interference directions) channel E (·) denotes expectation. Using binary signaling with rectangular pulse shapes minimizes the the minimizes shapes pulse rectangular with signaling binary Using expectation. (·) denotes rhgnl rqec dvso mlilxn (OFDM) multiplexing division frequency orthogonal ( PAR ). For a transmitted waveform waveform ). For atransmitted PA dpie precoding adaptive R = ma x E } { x ) ( t x t ) ( 2 2 (also known as Tomlinson–Harashima Tomlinson–Harashima as known (also x ( system, it is well known that the the that known well is it system, t ), the PAR is defined as PAR), the defined is ­appropriate for transmitted ­transmitted transmitter ­transmitter 51 - - - Downloaded By: 10.3.98.104 At: 10:19 25 Sep 2021; For: 9781439817247, chapter3, 10.1201/b12494-5 tral density and decreases the amount and density of decreases tral interference into ofa that system occupies part narrowband the powerspec the lowers signal the by occupied bandwidth the power, increasing average fixed assuming example, For shaping. pulse by and assignments slot) time perhaps (and frequency through controlled be can interference Cochannel signal. desired the as band frequency same the to assigned ­transmitters earlier, to addition described In radio. interference digital with adjacent-channel associated impairments channel primary the of one is Interference Tolerance to Interference 3.6.5 threshold). and filter (low-pass strategy detection simple If s 52 Third Generation Partnership Project 2 (3GPP2) 2 Project Partnership Generation Third the from evolution an is CDMA2000 CDMA2000 3.7.2 duration chip the where factor roll-off with pulse cosine raised asquare-root is modes as part of (3GPP) Project by developed the GSM standard European evolution the from is an which W-CDMA refers to the part of radio transmission Multiple (W-CDMA) Division Code Access Wideband 3.7.1 systems. cellular mobile for digital standards existing in used shapes pulse baseband of description brief a with chapter this Weconclude Examples 3.7 rate. error a target power for transmitted in reduction further enables a coding with Power-efficient combined modulation over density, a the power band and wide decreases frequency less hence the visible. spectral signal makes energy pulse the spreading since applications these in attractive are waveforms tion).Spread-spectrum detec of (low probability difficult be must place taking is or notthat communications whether is determining applications, some in requirement, additional An intercept). of probability (low conversations of user privacy the to guarantee is applications, military as well as for commercial, most A requirement The naturebroadcast ofeavesdroppingmakeseasier wirelessthan channels generally for wired Security and Privacy 3.6.6 service. either disrupting without signals on top of narrowband to be overlaid signals wideband therefore,enables spreading, bandwidth Sufficient bandwidth. available ufficient bandwidth is available, the cost can be reduced by using a rectangular pulse shape with a with shape pulse rectangular a using by reduced be can cost the available, is bandwidth ­ufficient [8]. It operates in both both [8]. in Itoperates UMTS UMTS Terrestrial Radio Access (UTRA) T c t p

) ( =

(1/chiprate) = si n    α π frequency-division duplex (FDD) duplex frequency-division T North American Interim Standard-95 (IS-95) Standard-95 Interim American North t C = ) ( 4 1

0.24414 − α π T t C . The 3GPP2 standard has defined baseband filters for filters baseband defined has standard 3GPP2 The .    Universal Universal Mobile Telecommunications System (UMTS)    4 1 ohne interference cochannel + μ − s. π α    α = . filterThe transmit pulse-shaping for W-CDMA T t C 0.22. The chip impulse response is given by is response impulse chip The 0.22. T c t C os       2 Mobile Communications Handbook Communications Mobile    T t C ) ( 1 and and + α may be generated by other other by generated be may Third Generation Partnership Partnership ThirdGeneration    time-division duplex (TDD) duplex time-division developed by the the by developed channels. ­channels. - - ,

Downloaded By: 10.3.98.104 At: 10:19 25 Sep 2021; For: 9781439817247, chapter3, 10.1201/b12494-5 therefore the square root of root square the therefore factor bandwidth excess with filter cosine raised square-root a specified [11]. has and The OFDM, standard on 802.16based is IEEE interface air the 802.16 to refers and technology, access wireless broadband a is WiMAX (WiMAX) for Microwave Access Worldwide Interoperability 3.7.4 of duration pulse toarectangular degenerates vanishing of case the In waveform. the transmitted reduces the of transition sidelobes this spectral Smoothing symbols. consecutive the between overlap small a creates This where by [10]:given is function time-windowing The modulation. to prior I- the signals filter to Q-channel used and be may that method pro has windowing 802.11 time-domain IEEE a for parameters objectives. those informative and vided achieve to windowing used be may time-domain methods of filtering in combination WLANs frequency-domain Any for 10.* mask Reference of spectral 17.3.9.6 the and for 17.3.9.2 requirements Sections developed has group working 802.11 IEEE The (WLAN) Networks Area Local Wireless 3.7.3 where factor scale a Specifically, filter is given by a scaled and delayedgiven is versionfilter by scaled a thecoefficients of where given by response frequency havefilters a normalized [9]. baseband These for CDMA2000 the In-phase (I-channel) (Q-channel) and prior signals Quadrature-phase to Shaping Pulse and Signaling Baseband *

Reference 10 was created by merging 802.11a, b, d, e, g, h, i, and j amendments with the base 802.11 standard and and standard 802.11 base the function.) windowing with amendments j this and references also 802.11n i, amendment h, 802.11-2007.high-throughput IEEE (The g, standard base e, current to the renamed d, b, 802.11a, merging by created was 10 Reference T δ T 1 s TR

= = (approximately 100

1.5 203.451 dB, dB, t w α = α δ T ns is one quarter of the duration of the CDMA2000 chip. CDMA2000 of the duration of the one is quarter ns 2 ) (

= 0.25 for the I- and Q-channels prior to modulation. The filter transfer function is function transfer filter The modulation. to prior Q-channels I- and for the 0.25

40 =          κ dB, dB, 2 1 si si and delay delay and , H ( n ( n fo f S ( 2 2 ns) is selected to OFDMbetween consecutive smooth transitions f ) ( f p / r      

) in Equation 3.18, Equation where ) in T = π π 2 2 TR

590 = 5 0 5 0 MS ( . ) .      τ t T t T < ≤ − ± kHz, and and kHz, are selected to minimize the mean squared error squared mean the tominimize selected are s E δ δ T t − − + 1 2 − = , , for / ∑ k T T in in ∞ = TR 0 − T T T ( [ th th τ κ . f T    ) ) s for / s e p e

TR , = kT TR

/ hopban 740 assban 2 s    < − , kHz. The impulse response response impulse The kHz. TR T ( ) is the OFDM symbol period. OFDM symbol the is d / / d − T T 2 2 0 k h f f ≤ − h  ( T t k )] TR < ) in Table) in 2.1.3.1.13.1-1 Reference in 9. f f 2 / s 2 TR p T t T + < TR , the windowing function function windowing the , T TR //2 s ( t ) of the baseband ) baseband of the modulation ­modulation symbols. ­symbols. 53 - Downloaded By: 10.3.98.104 At: 10:19 25 Sep 2021; For: 9781439817247, chapter3, 10.1201/b12494-5 References pulse transmit rectangular a uses E-UTRA (SC-FDMA) uses LTE technology. (OFDMA) access radio FDD half-duplex same as well as the modes, TDD with and FDD both in operates which LTE, for interface air the 8specification. 3GPPthe Release in defined of UMTS, evolution LTE an is Long-Term Evolution (LTE) 3.7.5 54 J. G. Proakis and M. Salehi, Salehi, M. and Proakis J. G. Messerschmitt, advanced G. D. and more Lee For A. E. Barry, J. communications. see treatments digital communications on texts digital numerous any in of covered design therefore, is, the and to ­system fundamental is shaping pulse and signaling Baseband Reading Further 12. 11. 10. 9. 8. 7. 6. 5. 4. 3. 2. 1. radio transmission and reception 2010. 9), October (Release Group Radio Access Network; Universal Terrestrial Radio Access (E-UTRA); User Equipment (UE) Wireless Access Systems, May, 2009. Medium Access Control (MAC) and Physical Layer (PHY) Specifications, June, 2007. Spread Systems. Spectrum TIA/EIA/IS-2000.2-A, March 2000. (FDD and 9),September TDD) 2010. (Release Specification GroupRadio Access Network; UserEquipment (UE) radiotransmission and reception 138, 139,1971. interference. IEEE Commun. Mag ­combination transmit of partial sequences, tions and detection. Tech. Electronics GP S 611 950 21-0, r Gnrto Prnrhp rjc; ehia Specification Technical Project; Partnership Generation 3rd (2010-10), V9.5.0 36.101 TS 3GPP IEEE Standard 802.16 802.11 Standard IEEE for Standards cdma2000 for Standard Layer Physical Association. Industry Telecommunication 3GPP TS 25.101 V9.5.0 and 25.102 V9.2.0 (2010-09), 3rd Generation Partnership Project; Technical arithmetic. modulo employingautomatic equalizer Tomlinson,New M., Matched-transmissionMiyakawa,H., intersymbolandHarashima, channels forwith techniqueH. come. has time whose idea An transmission: data for modulation Multicarrier C., A. J. Bingham, optimum by ratio power peak-to-average reduced with OFDM B., J. Huber, and H. S. Muller, Kalet, I. and Saltzberg, B. QAM R., transmission through a companding channel—Signal constella Kretzmer,Generalization techniqueof a R., for dataE. communication. binary Transmission.data high-speed for technique duobinary The A., Lender, radio-access for the downlink and and downlink the for radio-access , COM-14(Feb.), 67,68,1966. on the uplink. Unlike UTRA, which uses a square-root raised cosine pulse shaping filter, filter, shaping pulse cosine raised square-root a uses which UTRA, Unlike uplink. the on , 82(March), 214–218,1963. IEEE Trans. on Commun. IEEE Trans. on Comm. ., 28(May), 5–14,1990. TM Digital Communications Digital TM , Local and metropolitan area networks, Part 16: Air Interface for Broadband -2007, Local and metropolitan area networks, Part 11: Wireless LAN LAN Wireless 11: Part networks, area metropolitan and Local -2007, , COM-20(Aug.), 774–780,1972. shaping filter filter [12]. ­shaping , 42(2–4),417–429,1994. Electronic Letters Single-Carrier Frequency Division Multiple Access Access Multiple Division Frequency Single-Carrier , McGraw-Hill 2008. , McGraw-Hill Orthogonal Frequency Division Multiple Access Access Multiple Division Frequency Orthogonal Digital Communication Digital Mobile Communications Handbook Communications Mobile , 33(5),368-369,2002. Evolved UTRA (E-UTRA) UTRA Evolved Electron. Lett.Electron. AIEE Trans. on Comm. on Trans. AIEE IEEE Trans.IEEE Comm. , Kluwer 2004, and and 2004, Kluwer , , 7(March), , is is -